Entry requirements

120 credits including 90 credits in Mathematics. Basic Topology and Real Analysis recommended. Proficiency in English equivalent to the Swedish upper secondary course English 6.

Learning outcomes

On completion of the course, the student should be able to:

• give an account of central concepts and definitions in differential topology;
• state Sard's theorem and some of its applications;
• define and compute mapping degree and intersection number of two submanifolds;
• define index of a vector field and state the Poincaré-Hopf theorem;
• define Morse function and outline a proof of existence;
• state the classification of one- and two-dimensional manifolds.

Content

Manifolds, submanifolds, smooth maps, tangent space, tangent bundles. The constant rank theorem. Immersions and embeddings. Submersions. Transversality. Sard's theorem. Brouwer's fix point theorem. Intersection theory modulo 2. Jordan-Brouwer's separation theorem. Orientation. Oriented intersection theory. Lefschetz's fix point theorem. The degree of a map. Hopf's degree theorem. Vector fields on manifolds and Poincaré-Hopf's theorem. Morse functions. Classification of one- and two-dimensional manifolds.

Instruction

Lectures and problem solving sessions.

Assessment

Compulsory assignments. Oral examination may be given.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.